Average Error: 29.9 → 7.8
Time: 11.3s
Precision: 64
Internal Precision: 128
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.9

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]

  2. Initial simplification7.8

    \[\leadsto \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]

  3. Final simplification7.8

    \[\leadsto \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))

Details

Time bar (total: 9.1s)Debug log

start33.0ms

Algorithm
intervals

setup226.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 4.5b

localize13.0ms

Local error

Found 2 expressions with local error:

4.6b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

3.0ms
(* (/ b a) (/ b a))
1.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

series26.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

14.0ms
(* (/ b a) (/ b a))
12.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.1s

Counts
25 → 47
Calls

25 calls. Slowest were:

803.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
39.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))
36.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune520.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 4.5b

localize30.0ms

Local error

Found 3 expressions with local error:

7.1b
(* (* (/ b a) b) (/ 1 a))
4.6b
(sqrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a)))))
0.2b
(* (/ b a) b)

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
3 → 51
Calls

3 calls. Slowest were:

13.0ms
(* (* (/ b a) b) (/ 1 a))
3.0ms
(* (/ b a) b)
2.0ms
(sqrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a)))))

series40.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

14.0ms
(* (* (/ b a) b) (/ 1 a))
13.0ms
(sqrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a)))))
12.0ms
(* (/ b a) b)

simplify2.6s

Counts
27 → 60
Calls

27 calls. Slowest were:

862.0ms
(* (* (* (* (/ b a) b) (* (/ b a) b)) (* (/ b a) b)) (* (* (/ 1 a) (/ 1 a)) (/ 1 a)))
615.0ms
(sqrt (sqrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a))))))
303.0ms
(sqrt (* (cbrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a))))) (cbrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a)))))))

prune627.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 4.5b

localize8.0ms

Local error

Found 4 expressions with local error:

4.6b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
1.5b
(log (exp (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))
0.4b
(* (/ b a) (/ b a))
0.0b
(exp (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

3.0ms
(* (/ b a) (/ b a))
3.0ms
(log (exp (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))
2.0ms
(exp (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))

series85.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

44.0ms
(exp (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))
14.0ms
(* (/ b a) (/ b a))
13.0ms
(log (exp (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))
13.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify2.6s

Counts
48 → 89
Calls

48 calls. Slowest were:

1.0s
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
73.0ms
(exp (sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a))))))))
70.0ms
(exp (* (cbrt (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))) (cbrt (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))))

prune1.1s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 4.5b

regimes77.0ms

Accuracy

0% (0.0b remaining)

Error of 7.8b against oracle of 7.8b and baseline of 7.8b

bsearch2.0ms